// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2006-2009 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_PARTIALLU_H
#define EIGEN_PARTIALLU_H

namespace Eigen {

namespace internal {
    template <typename _MatrixType> struct traits<PartialPivLU<_MatrixType>> : traits<_MatrixType>
    {
        typedef MatrixXpr XprKind;
        typedef SolverStorage StorageKind;
        typedef int StorageIndex;
        typedef traits<_MatrixType> BaseTraits;
        enum
        {
            Flags = BaseTraits::Flags & RowMajorBit,
            CoeffReadCost = Dynamic
        };
    };

    template <typename T, typename Derived> struct enable_if_ref;
    // {
    //   typedef Derived type;
    // };

    template <typename T, typename Derived> struct enable_if_ref<Ref<T>, Derived>
    {
        typedef Derived type;
    };

}  // end namespace internal

/** \ingroup LU_Module
  *
  * \class PartialPivLU
  *
  * \brief LU decomposition of a matrix with partial pivoting, and related features
  *
  * \tparam _MatrixType the type of the matrix of which we are computing the LU decomposition
  *
  * This class represents a LU decomposition of a \b square \b invertible matrix, with partial pivoting: the matrix A
  * is decomposed as A = PLU where L is unit-lower-triangular, U is upper-triangular, and P
  * is a permutation matrix.
  *
  * Typically, partial pivoting LU decomposition is only considered numerically stable for square invertible
  * matrices. Thus LAPACK's dgesv and dgesvx require the matrix to be square and invertible. The present class
  * does the same. It will assert that the matrix is square, but it won't (actually it can't) check that the
  * matrix is invertible: it is your task to check that you only use this decomposition on invertible matrices.
  *
  * The guaranteed safe alternative, working for all matrices, is the full pivoting LU decomposition, provided
  * by class FullPivLU.
  *
  * This is \b not a rank-revealing LU decomposition. Many features are intentionally absent from this class,
  * such as rank computation. If you need these features, use class FullPivLU.
  *
  * This LU decomposition is suitable to invert invertible matrices. It is what MatrixBase::inverse() uses
  * in the general case.
  * On the other hand, it is \b not suitable to determine whether a given matrix is invertible.
  *
  * The data of the LU decomposition can be directly accessed through the methods matrixLU(), permutationP().
  *
  * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
  *
  * \sa MatrixBase::partialPivLu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse(), class FullPivLU
  */
template <typename _MatrixType> class PartialPivLU : public SolverBase<PartialPivLU<_MatrixType>>
{
public:
    typedef _MatrixType MatrixType;
    typedef SolverBase<PartialPivLU> Base;
    friend class SolverBase<PartialPivLU>;

    EIGEN_GENERIC_PUBLIC_INTERFACE(PartialPivLU)
    enum
    {
        MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
        MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
    };
    typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationType;
    typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> TranspositionType;
    typedef typename MatrixType::PlainObject PlainObject;

    /**
      * \brief Default Constructor.
      *
      * The default constructor is useful in cases in which the user intends to
      * perform decompositions via PartialPivLU::compute(const MatrixType&).
      */
    PartialPivLU();

    /** \brief Default Constructor with memory preallocation
      *
      * Like the default constructor but with preallocation of the internal data
      * according to the specified problem \a size.
      * \sa PartialPivLU()
      */
    explicit PartialPivLU(Index size);

    /** Constructor.
      *
      * \param matrix the matrix of which to compute the LU decomposition.
      *
      * \warning The matrix should have full rank (e.g. if it's square, it should be invertible).
      * If you need to deal with non-full rank, use class FullPivLU instead.
      */
    template <typename InputType> explicit PartialPivLU(const EigenBase<InputType>& matrix);

    /** Constructor for \link InplaceDecomposition inplace decomposition \endlink
      *
      * \param matrix the matrix of which to compute the LU decomposition.
      *
      * \warning The matrix should have full rank (e.g. if it's square, it should be invertible).
      * If you need to deal with non-full rank, use class FullPivLU instead.
      */
    template <typename InputType> explicit PartialPivLU(EigenBase<InputType>& matrix);

    template <typename InputType> PartialPivLU& compute(const EigenBase<InputType>& matrix)
    {
        m_lu = matrix.derived();
        compute();
        return *this;
    }

    /** \returns the LU decomposition matrix: the upper-triangular part is U, the
      * unit-lower-triangular part is L (at least for square matrices; in the non-square
      * case, special care is needed, see the documentation of class FullPivLU).
      *
      * \sa matrixL(), matrixU()
      */
    inline const MatrixType& matrixLU() const
    {
        eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
        return m_lu;
    }

    /** \returns the permutation matrix P.
      */
    inline const PermutationType& permutationP() const
    {
        eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
        return m_p;
    }

#ifdef EIGEN_PARSED_BY_DOXYGEN
    /** This method returns the solution x to the equation Ax=b, where A is the matrix of which
      * *this is the LU decomposition.
      *
      * \param b the right-hand-side of the equation to solve. Can be a vector or a matrix,
      *          the only requirement in order for the equation to make sense is that
      *          b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition.
      *
      * \returns the solution.
      *
      * Example: \include PartialPivLU_solve.cpp
      * Output: \verbinclude PartialPivLU_solve.out
      *
      * Since this PartialPivLU class assumes anyway that the matrix A is invertible, the solution
      * theoretically exists and is unique regardless of b.
      *
      * \sa TriangularView::solve(), inverse(), computeInverse()
      */
    template <typename Rhs> inline const Solve<PartialPivLU, Rhs> solve(const MatrixBase<Rhs>& b) const;
#endif

    /** \returns an estimate of the reciprocal condition number of the matrix of which \c *this is
        the LU decomposition.
      */
    inline RealScalar rcond() const
    {
        eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
        return internal::rcond_estimate_helper(m_l1_norm, *this);
    }

    /** \returns the inverse of the matrix of which *this is the LU decomposition.
      *
      * \warning The matrix being decomposed here is assumed to be invertible. If you need to check for
      *          invertibility, use class FullPivLU instead.
      *
      * \sa MatrixBase::inverse(), LU::inverse()
      */
    inline const Inverse<PartialPivLU> inverse() const
    {
        eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
        return Inverse<PartialPivLU>(*this);
    }

    /** \returns the determinant of the matrix of which
      * *this is the LU decomposition. It has only linear complexity
      * (that is, O(n) where n is the dimension of the square matrix)
      * as the LU decomposition has already been computed.
      *
      * \note For fixed-size matrices of size up to 4, MatrixBase::determinant() offers
      *       optimized paths.
      *
      * \warning a determinant can be very big or small, so for matrices
      * of large enough dimension, there is a risk of overflow/underflow.
      *
      * \sa MatrixBase::determinant()
      */
    Scalar determinant() const;

    MatrixType reconstructedMatrix() const;

    EIGEN_CONSTEXPR inline Index rows() const EIGEN_NOEXCEPT { return m_lu.rows(); }
    EIGEN_CONSTEXPR inline Index cols() const EIGEN_NOEXCEPT { return m_lu.cols(); }

#ifndef EIGEN_PARSED_BY_DOXYGEN
    template <typename RhsType, typename DstType> EIGEN_DEVICE_FUNC void _solve_impl(const RhsType& rhs, DstType& dst) const
    {
        /* The decomposition PA = LU can be rewritten as A = P^{-1} L U.
      * So we proceed as follows:
      * Step 1: compute c = Pb.
      * Step 2: replace c by the solution x to Lx = c.
      * Step 3: replace c by the solution x to Ux = c.
      */

        // Step 1
        dst = permutationP() * rhs;

        // Step 2
        m_lu.template triangularView<UnitLower>().solveInPlace(dst);

        // Step 3
        m_lu.template triangularView<Upper>().solveInPlace(dst);
    }

    template <bool Conjugate, typename RhsType, typename DstType> EIGEN_DEVICE_FUNC void _solve_impl_transposed(const RhsType& rhs, DstType& dst) const
    {
        /* The decomposition PA = LU can be rewritten as A^T = U^T L^T P.
      * So we proceed as follows:
      * Step 1: compute c as the solution to L^T c = b
      * Step 2: replace c by the solution x to U^T x = c.
      * Step 3: update  c = P^-1 c.
      */

        eigen_assert(rhs.rows() == m_lu.cols());

        // Step 1
        dst = m_lu.template triangularView<Upper>().transpose().template conjugateIf<Conjugate>().solve(rhs);
        // Step 2
        m_lu.template triangularView<UnitLower>().transpose().template conjugateIf<Conjugate>().solveInPlace(dst);
        // Step 3
        dst = permutationP().transpose() * dst;
    }
#endif

protected:
    static void check_template_parameters() { EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar); }

    void compute();

    MatrixType m_lu;
    PermutationType m_p;
    TranspositionType m_rowsTranspositions;
    RealScalar m_l1_norm;
    signed char m_det_p;
    bool m_isInitialized;
};

template <typename MatrixType>
PartialPivLU<MatrixType>::PartialPivLU() : m_lu(), m_p(), m_rowsTranspositions(), m_l1_norm(0), m_det_p(0), m_isInitialized(false)
{
}

template <typename MatrixType>
PartialPivLU<MatrixType>::PartialPivLU(Index size) : m_lu(size, size), m_p(size), m_rowsTranspositions(size), m_l1_norm(0), m_det_p(0), m_isInitialized(false)
{
}

template <typename MatrixType>
template <typename InputType>
PartialPivLU<MatrixType>::PartialPivLU(const EigenBase<InputType>& matrix)
    : m_lu(matrix.rows(), matrix.cols()), m_p(matrix.rows()), m_rowsTranspositions(matrix.rows()), m_l1_norm(0), m_det_p(0), m_isInitialized(false)
{
    compute(matrix.derived());
}

template <typename MatrixType>
template <typename InputType>
PartialPivLU<MatrixType>::PartialPivLU(EigenBase<InputType>& matrix)
    : m_lu(matrix.derived()), m_p(matrix.rows()), m_rowsTranspositions(matrix.rows()), m_l1_norm(0), m_det_p(0), m_isInitialized(false)
{
    compute();
}

namespace internal {

    /** \internal This is the blocked version of fullpivlu_unblocked() */
    template <typename Scalar, int StorageOrder, typename PivIndex, int SizeAtCompileTime = Dynamic> struct partial_lu_impl
    {
        static const int UnBlockedBound = 16;
        static const bool UnBlockedAtCompileTime = SizeAtCompileTime != Dynamic && SizeAtCompileTime <= UnBlockedBound;
        static const int ActualSizeAtCompileTime = UnBlockedAtCompileTime ? SizeAtCompileTime : Dynamic;
        // Remaining rows and columns at compile-time:
        static const int RRows = SizeAtCompileTime == 2 ? 1 : Dynamic;
        static const int RCols = SizeAtCompileTime == 2 ? 1 : Dynamic;
        typedef Matrix<Scalar, ActualSizeAtCompileTime, ActualSizeAtCompileTime, StorageOrder> MatrixType;
        typedef Ref<MatrixType> MatrixTypeRef;
        typedef Ref<Matrix<Scalar, Dynamic, Dynamic, StorageOrder>> BlockType;
        typedef typename MatrixType::RealScalar RealScalar;

        /** \internal performs the LU decomposition in-place of the matrix \a lu
    * using an unblocked algorithm.
    *
    * In addition, this function returns the row transpositions in the
    * vector \a row_transpositions which must have a size equal to the number
    * of columns of the matrix \a lu, and an integer \a nb_transpositions
    * which returns the actual number of transpositions.
    *
    * \returns The index of the first pivot which is exactly zero if any, or a negative number otherwise.
    */
        static Index unblocked_lu(MatrixTypeRef& lu, PivIndex* row_transpositions, PivIndex& nb_transpositions)
        {
            typedef scalar_score_coeff_op<Scalar> Scoring;
            typedef typename Scoring::result_type Score;
            const Index rows = lu.rows();
            const Index cols = lu.cols();
            const Index size = (std::min)(rows, cols);
            // For small compile-time matrices it is worth processing the last row separately:
            //  speedup: +100% for 2x2, +10% for others.
            const Index endk = UnBlockedAtCompileTime ? size - 1 : size;
            nb_transpositions = 0;
            Index first_zero_pivot = -1;
            for (Index k = 0; k < endk; ++k)
            {
                int rrows = internal::convert_index<int>(rows - k - 1);
                int rcols = internal::convert_index<int>(cols - k - 1);

                Index row_of_biggest_in_col;
                Score biggest_in_corner = lu.col(k).tail(rows - k).unaryExpr(Scoring()).maxCoeff(&row_of_biggest_in_col);
                row_of_biggest_in_col += k;

                row_transpositions[k] = PivIndex(row_of_biggest_in_col);

                if (biggest_in_corner != Score(0))
                {
                    if (k != row_of_biggest_in_col)
                    {
                        lu.row(k).swap(lu.row(row_of_biggest_in_col));
                        ++nb_transpositions;
                    }

                    lu.col(k).tail(fix<RRows>(rrows)) /= lu.coeff(k, k);
                }
                else if (first_zero_pivot == -1)
                {
                    // the pivot is exactly zero, we record the index of the first pivot which is exactly 0,
                    // and continue the factorization such we still have A = PLU
                    first_zero_pivot = k;
                }

                if (k < rows - 1)
                    lu.bottomRightCorner(fix<RRows>(rrows), fix<RCols>(rcols)).noalias() -=
                        lu.col(k).tail(fix<RRows>(rrows)) * lu.row(k).tail(fix<RCols>(rcols));
            }

            // special handling of the last entry
            if (UnBlockedAtCompileTime)
            {
                Index k = endk;
                row_transpositions[k] = PivIndex(k);
                if (Scoring()(lu(k, k)) == Score(0) && first_zero_pivot == -1)
                    first_zero_pivot = k;
            }

            return first_zero_pivot;
        }

        /** \internal performs the LU decomposition in-place of the matrix represented
    * by the variables \a rows, \a cols, \a lu_data, and \a lu_stride using a
    * recursive, blocked algorithm.
    *
    * In addition, this function returns the row transpositions in the
    * vector \a row_transpositions which must have a size equal to the number
    * of columns of the matrix \a lu, and an integer \a nb_transpositions
    * which returns the actual number of transpositions.
    *
    * \returns The index of the first pivot which is exactly zero if any, or a negative number otherwise.
    *
    * \note This very low level interface using pointers, etc. is to:
    *   1 - reduce the number of instantiations to the strict minimum
    *   2 - avoid infinite recursion of the instantiations with Block<Block<Block<...> > >
    */
        static Index
        blocked_lu(Index rows, Index cols, Scalar* lu_data, Index luStride, PivIndex* row_transpositions, PivIndex& nb_transpositions, Index maxBlockSize = 256)
        {
            MatrixTypeRef lu = MatrixType::Map(lu_data, rows, cols, OuterStride<>(luStride));

            const Index size = (std::min)(rows, cols);

            // if the matrix is too small, no blocking:
            if (UnBlockedAtCompileTime || size <= UnBlockedBound)
            {
                return unblocked_lu(lu, row_transpositions, nb_transpositions);
            }

            // automatically adjust the number of subdivisions to the size
            // of the matrix so that there is enough sub blocks:
            Index blockSize;
            {
                blockSize = size / 8;
                blockSize = (blockSize / 16) * 16;
                blockSize = (std::min)((std::max)(blockSize, Index(8)), maxBlockSize);
            }

            nb_transpositions = 0;
            Index first_zero_pivot = -1;
            for (Index k = 0; k < size; k += blockSize)
            {
                Index bs = (std::min)(size - k, blockSize);  // actual size of the block
                Index trows = rows - k - bs;                 // trailing rows
                Index tsize = size - k - bs;                 // trailing size

                // partition the matrix:
                //                          A00 | A01 | A02
                // lu  = A_0 | A_1 | A_2 =  A10 | A11 | A12
                //                          A20 | A21 | A22
                BlockType A_0 = lu.block(0, 0, rows, k);
                BlockType A_2 = lu.block(0, k + bs, rows, tsize);
                BlockType A11 = lu.block(k, k, bs, bs);
                BlockType A12 = lu.block(k, k + bs, bs, tsize);
                BlockType A21 = lu.block(k + bs, k, trows, bs);
                BlockType A22 = lu.block(k + bs, k + bs, trows, tsize);

                PivIndex nb_transpositions_in_panel;
                // recursively call the blocked LU algorithm on [A11^T A21^T]^T
                // with a very small blocking size:
                Index ret = blocked_lu(trows + bs, bs, &lu.coeffRef(k, k), luStride, row_transpositions + k, nb_transpositions_in_panel, 16);
                if (ret >= 0 && first_zero_pivot == -1)
                    first_zero_pivot = k + ret;

                nb_transpositions += nb_transpositions_in_panel;
                // update permutations and apply them to A_0
                for (Index i = k; i < k + bs; ++i)
                {
                    Index piv = (row_transpositions[i] += internal::convert_index<PivIndex>(k));
                    A_0.row(i).swap(A_0.row(piv));
                }

                if (trows)
                {
                    // apply permutations to A_2
                    for (Index i = k; i < k + bs; ++i) A_2.row(i).swap(A_2.row(row_transpositions[i]));

                    // A12 = A11^-1 A12
                    A11.template triangularView<UnitLower>().solveInPlace(A12);

                    A22.noalias() -= A21 * A12;
                }
            }
            return first_zero_pivot;
        }
    };

    /** \internal performs the LU decomposition with partial pivoting in-place.
  */
    template <typename MatrixType, typename TranspositionType>
    void partial_lu_inplace(MatrixType& lu, TranspositionType& row_transpositions, typename TranspositionType::StorageIndex& nb_transpositions)
    {
        // Special-case of zero matrix.
        if (lu.rows() == 0 || lu.cols() == 0)
        {
            nb_transpositions = 0;
            return;
        }
        eigen_assert(lu.cols() == row_transpositions.size());
        eigen_assert(row_transpositions.size() < 2 || (&row_transpositions.coeffRef(1) - &row_transpositions.coeffRef(0)) == 1);

        partial_lu_impl<typename MatrixType::Scalar,
                        MatrixType::Flags & RowMajorBit ? RowMajor : ColMajor,
                        typename TranspositionType::StorageIndex,
                        EIGEN_SIZE_MIN_PREFER_FIXED(MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime)>::blocked_lu(lu.rows(),
                                                                                                                               lu.cols(),
                                                                                                                               &lu.coeffRef(0, 0),
                                                                                                                               lu.outerStride(),
                                                                                                                               &row_transpositions.coeffRef(0),
                                                                                                                               nb_transpositions);
    }

}  // end namespace internal

template <typename MatrixType> void PartialPivLU<MatrixType>::compute()
{
    check_template_parameters();

    // the row permutation is stored as int indices, so just to be sure:
    eigen_assert(m_lu.rows() < NumTraits<int>::highest());

    if (m_lu.cols() > 0)
        m_l1_norm = m_lu.cwiseAbs().colwise().sum().maxCoeff();
    else
        m_l1_norm = RealScalar(0);

    eigen_assert(m_lu.rows() == m_lu.cols() && "PartialPivLU is only for square (and moreover invertible) matrices");
    const Index size = m_lu.rows();

    m_rowsTranspositions.resize(size);

    typename TranspositionType::StorageIndex nb_transpositions;
    internal::partial_lu_inplace(m_lu, m_rowsTranspositions, nb_transpositions);
    m_det_p = (nb_transpositions % 2) ? -1 : 1;

    m_p = m_rowsTranspositions;

    m_isInitialized = true;
}

template <typename MatrixType> typename PartialPivLU<MatrixType>::Scalar PartialPivLU<MatrixType>::determinant() const
{
    eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
    return Scalar(m_det_p) * m_lu.diagonal().prod();
}

/** \returns the matrix represented by the decomposition,
 * i.e., it returns the product: P^{-1} L U.
 * This function is provided for debug purpose. */
template <typename MatrixType> MatrixType PartialPivLU<MatrixType>::reconstructedMatrix() const
{
    eigen_assert(m_isInitialized && "LU is not initialized.");
    // LU
    MatrixType res = m_lu.template triangularView<UnitLower>().toDenseMatrix() * m_lu.template triangularView<Upper>();

    // P^{-1}(LU)
    res = m_p.inverse() * res;

    return res;
}

/***** Implementation details *****************************************************/

namespace internal {

    /***** Implementation of inverse() *****************************************************/
    template <typename DstXprType, typename MatrixType>
    struct Assignment<DstXprType,
                      Inverse<PartialPivLU<MatrixType>>,
                      internal::assign_op<typename DstXprType::Scalar, typename PartialPivLU<MatrixType>::Scalar>,
                      Dense2Dense>
    {
        typedef PartialPivLU<MatrixType> LuType;
        typedef Inverse<LuType> SrcXprType;
        static void run(DstXprType& dst, const SrcXprType& src, const internal::assign_op<typename DstXprType::Scalar, typename LuType::Scalar>&)
        {
            dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
        }
    };
}  // end namespace internal

/******** MatrixBase methods *******/

/** \lu_module
  *
  * \return the partial-pivoting LU decomposition of \c *this.
  *
  * \sa class PartialPivLU
  */
template <typename Derived> inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject> MatrixBase<Derived>::partialPivLu() const
{
    return PartialPivLU<PlainObject>(eval());
}

/** \lu_module
  *
  * Synonym of partialPivLu().
  *
  * \return the partial-pivoting LU decomposition of \c *this.
  *
  * \sa class PartialPivLU
  */
template <typename Derived> inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject> MatrixBase<Derived>::lu() const
{
    return PartialPivLU<PlainObject>(eval());
}

}  // end namespace Eigen

#endif  // EIGEN_PARTIALLU_H
